Two vertices of a right triangle have coordinates (20, 15) and (20, 27) . the segment that connects these points is a leg of the triangle. which coordinate pair for the third vertex would create a right triangle?

The coordinate pairs for third vertex are not posted.  So the answer is limited to conditions on which coordinate pairs of the third vertex must satisfy.

The two given points form a vertical segment.  So any horizontal segment that ends at one of the points will form a right triangle.

Since the vertical segment has a length of (27-15)=12, the following example pairs will give the popular (5-12-13) right triangle, i.e. the other leg is horizontal and having a length of 5.

Leg ending at (20,15):  (15,15), or (25,15)
Leg ending at (20,27):  (15,27), or (25,27)

Any vertex points satisfying (x,15) or (x,27) are valid candidates for all x ≠ 20.